Find the Derivative - d/dx x^(sin(x))

Problem

d(xsin(x))/d(x)

Solution

1. Set the expression equal to y to use logarithmic differentiation.

y=xsin(x)

2. Take the natural log of both sides to move the exponent.

ln(y)=ln(xsin(x))

3. Apply the power rule for logarithms to simplify the right side.

ln(y)=sin(x)*ln(x)

4. Differentiate implicitly both sides with respect to x

d(ln(y))/d(x)=(d(sin(x))*ln(x))/d(x)

5. Apply the chain rule to the left side and the product rule to the right side.

1/yd(y)/d(x)=cos(x)*ln(x)+sin(x)1/x

6. Solve for the derivative by multiplying both sides by y

d(y)/d(x)=y*(cos(x)*ln(x)+sin(x)/x)

7. Substitute the original expression for y back into the equation.

d(y)/d(x)=xsin(x)*(cos(x)*ln(x)+sin(x)/x)

Final Answer

d(xsin(x))/d(x)=xsin(x)*(cos(x)*ln(x)+sin(x)/x)

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